Armature not working after linking new mesh data
Logistically speaking, If it is now a single mesh with verts associated with vertex groups with the appropriate bone names from the armature that it is already parented to, it should work. 

I just ran some tests on a rig that I have. I added a cube in Object mode, joined it to the rigged mesh, in Edit mode I copied the vertex groups associated with a vert in the original mesh, went into pose mode and it worked without a hitch.

If you upload the .blend I'll take a peek.